C₂²×C₄×#731 #13047489

order	1296
nilpotency class	3
isoclinism	isoclinic class of AΣL(1,𝔽₂₇)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₂²×C₄×#731

Subgroups

	subgroup	G/H
center	C₂²×C₃×C₄	H(3,𝔽₃)
commutator	C₃²	C₂²×C₃²×C₄
Frattini, cosocle	C₂×C₃²	C₂³×C₃²
socle	C₂³×C₃	C₂×H(3,𝔽₃)
prime radical	C₂³×H(3,𝔽₃)	C₂×C₃
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃²×C₄ ◁ G

378 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 432

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2)(9,10,13)(11,17,21)(12,16,22)(14,23,19)(15,26,20)(18,28,31)(24,34,35)(25,33,30)(27,32,29);(3,4)(9,11,18,12,19,29,20,30,35)(10,14,24,16,25,28,15,17,27)(13,21,31,22,23,32,26,33,34);(5,6,7,8)

Conjugacy classes & character table

272 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 1207);